Spectral And High-Order Time Discretization Methods With Applications

Partial differential equations (PDE) can rarely be solved analytically, numer-ical approximations of solutions (numerical solutions) play a big role in applied science. Many time-dependent semilinear parabolic:PDEs combine low-order nonlinear terms with higher-order linear terms (where parabolic means that, the linear part is smoothing, and then semilinear means that the nonlinearity con-tains less derivatives than the linear part), such as Burgers equation, Allen-Cahn equation, Cahn-Hilliard equation, Fisher-KPP equation, Fitzhugh-Naguno equation, Gray-Scott equation, Hodgkin-Huxley equation, Korteweg-de Vries equation. Kuramoto-Sivashinsky equation, Navier-Stokes equation, nonlinear Schrodinger equation. To obtain accurate numerical solutions of such problem-s, it is desirable to use high-order approximations in space and time. Yet the application of finite difference, finite element methods discretize the spatial vari-ables, because of the difficulties introduced by the combination of nonlinearity and stiffness, most computations heretofore have been limited to second order in time.This dissertation studies some PDEs models with spectral methods and high-order time discretization methods, including Burgers equation, Korteweg-de Vries equation, Kuramoto-Sivashinsky equation, Allen-Cahn equation, Cahn-Hilliard equation, and nonlinear Schrodinger equation. For elliptic equations, for instance Helmholtz equation, biharmonic equation, spectral methods using less nodes and proper orthogonal polynomials which compact combination of Cheby-shev polynomials, or Legendre polynomials as basis functions, will have higher order spectral accuracy [,,] by solving a banded linear system compared with finite difference, finite element methods for solving equations on the regu-lar geometry domain as long as the problems posses sufficient smoothness. For unbounded domain, finite difference, finite element methods usually truncate the infinite domain into finite domain, and introduce artificial boundary condition-s. However, it is not necessary to truncate for spectral methods in unbounded domains. Laguerre spectral methods [] in a semi-infinite interval and Hermite spectral methods [] in an infinite interval are applied to solve differential equa-tions in unbounded domains. For periodic problems, whatever one dimension or higher dimension, it is easy to implement using Fourier spectral method. The disadvantage of spectral is that the solutions for equations are sufficient smooth-ness, therefore it is most applied to numerical examples, can’t popular in specific applied problemsAlthough there are many excellent books on spectral methods at home and abroad [,,,,,,,], spectral methods didn’t receive much attention at home. Classical spectral methods included three types of spectral schemes, namely, the Galerkin, collocation, and tau versions. The collocation method is the most popular form of the spectral methods among practitioner-s. It is simple, and very easy to implement, in particular for solving variable coefficients and nonlinear problems. The drawback of the collocation method is that the differentiation matrices are dense. Spectral Galerkin method is sim-ilar to finite element method using the variational form, isn’t hard to derive error analysis, but hard to implement. The reason is that, we need to transform between the physical space and the spectral space. Trefethen’s book [] ex-plained the essentials of spectral collocation methods with the aid of40short MATLAB programs. We provide a sufficient background on the implementation and some MATLAB codes corresponding to algorithms for solving the period-ic problems using Fourier spectral method, the Robin boundary problems using Chebyshev collocation method, the second-order and fourth-order equations us-ing the Legendre-Galerkin methods. We adopt the form:theory algorithm code-example, so that the reads can understand the algorithms and codes. Especially the codes for the Legendre-Galerkin methods, they can be as templates to study the spectral-Galerkin methods. We recommend Jie Shen’s new book [] as a textbook to understand and study spectral methods.We usually adopt the method of lines (MoL)[] for numerically solving semilinear parabolic PDEs. apply spectral methods to discretize the spatial vari-ables and generates a large coupled system of ordinary differential equations (ODEs) in time. Existing methods for solving ODEs can be classified into two groups. The first group consists of intrinsically high-order discretization schemes. e.g., Runge-Kutta (RK) methods, linear multistep methods, etc. The second group consists of methods based on accelerating the convergence of low-order schemes through the use of Richardson extrapolation or deferred correction. For non-stiff problems, there exist extremely effective discretizations of order up to twelve or so. For stiff problems, implicit Runge-Kutta methods possess excellent stability properties, but are very expensive when high order accuracy is required. Implicit multistep algorithms can have very high order convergence, but tend to have relatively poor stability properties. Therefore, most practitioners recom-mend some form of Runge-Kutta method for orders up to five or so, and again turn to extrapolation when higher order accuracy is needed. These extrapola-tion methods, while effective, are stiff expensive, since they require computing a sequence of solutions on finer and finer grids. For PDEs with higher order spatial derivatives, especially when the coefficients in front of these higher or-der spatial derivative terms are not small, the spatial variables are discretized by spectral methods to generat.e a large coupled system of ordinary differential equa-tions in time. The explicit and local time discretization suffers from extremely small time step restriction for stability, so it is desired to construct a stabilized scheme. There exist three different time discretization techniques for solving the stiff ODEs resulting from spectral spatial discretization to PDEs with higher or-der spatial derivatives. These are the semi-implicit spectral deferred correction (SISDC) method, the additive Runge-Kutta (ARK) method and the exponential time differencing (ETD) method. In chapter3, we introduce some high-order time discretization schemes, including the fractional step methods used in mod-elling Hamiltonian dynamics, integrating factor (IF) method originated from the theory of ODEs, exponential time differencing (ETD) method, SISDC method and Krylov deferred correction (KDC) method.The dissertation’s focus is on how spectral methods and high-order time dis-cretization methods solve PDEs in physical models. We study the Korteweg-de Vries equation on shallow water surfaces. Kuramoto-Sivashinsky equation used in the study of a variety of reaction-diffusion systems, Allen-Calm and Cahn-Hilliard equations in materials science, and nonlinear Schrodinger equation in mathematical physics. Here we consider Allen-Cahn equation and the Cahn-Hilliard equation The Allen-Cahn equation was originally introduced in [] to describe the mo-tion of anti-phase boundaries in crystalline solids, while the Cahn-Hilliard e-quation was introduced by in [] to describe the complicated phase separation and coarsening phenomena in a solid. These equations have been used exten-sively in materials science as model for phase transition, pattern formation etc. and have been the subject of extensive mathematical and numerical studies (see, e.g.,[,,,] and the references therein). It is still a challenge to construct efficient high-order, unconditionally energy stable schemes which are robust with small ε. Stabilized semi-implicit spectral defect correction (SSISDC) methods are constructed for the time discretization of Allen-Cahn and Cahn-Hilliard e-quations in []. These methods are unconditionally stable, lead to simple linear system to solve at each iteration and can achieve high-order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the SSISDC methods for solving the Allen-Cahn and Cahn-Hilliard equations. While we only consider the Allen-Cahn and Cahn-Hilliard equations in this dissertation, it is clear that this approach can be applied to more general gradient flows, including the equations for thin film epitaxy (cf.[,,]) and anisotropic Cahn-Hilliard equations (cf.[,,]). More-over, this approach is an important component for developing high-order stable schemes for phase-field models of the multi-phase flows (of. for instance [] and the references therein).